Supersymmetry and the Dirac Equation...Euclidean 4-dimensional Dirac equation. We then find all shape invariant poten- tials in 2 dimensions for which the Dirac equation is exactly

ANNALS OF PHYSICS 187, 1-28 (1988)

Supersymmetry and the Dirac Equation

FRED CARPER

Theorelical Division. Los Alamos National Laboratory, Los Alamos. New Mexico 87545

AVINASH KHARE*

Department of Physics, University of Illinois, Chicago, Illinois 60680

R. MUSTO

Dipartimento di Scienze Fisiche, Vniversitci di Napoli, and INFN Sezione di Napoli, Napoli, Italy

AND

A. WIPF+

Theorelical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Received April 1, 1988

We discuss in detail two supersymmetries of the 4-dimensional Dirac operator D2 where 0 = B - ieA, namely the usual chiral supersymmetry and a separate complex supersymmetry. Using SUSY methods developed to categorize solvable potentials in l-dimensional quantum mechanics we systematically study the cases where the spectrum, eigenfunctions, and S-matrix of 0* can be obtained analytically. We relate these solutions to the solutions of the ordinary massive Dirac equation in external fields. We show that whenever a Schrijdinger equation for a potential V(x) is exactly solvable, then there always exists a corresponding static scalar field q(x) for which the Jackiw-Rebbi type (1 + 1 )-dimensional Dirac equation is exactly solvable with k’(x) and q(x) being related by V(x) = c?(x) + cp’(x). We also discuss and exploit the supersymmetry of the path integral representation for the fermion propagator in an external held. ‘I;: 1988 Academx Press, Inc.

I. INTRODUCTION

Recently there has been a renewed interest in understanding the zero modes and complete spectrum of the square of the Dirac operatbr for a Euclidean fermionic theory interacting with background gauge fields. This operator can be considered

* Permanent address: Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India. + Present address: Max Planck Institut, Fohringer Ring 6, 8000 Munchen 40, West Germany.

OOO3-4916/88 $7.50 Copyright 0 1988 by Academic Press, Inc.

All rights of reproduction in any fom~ reserved.

2 COOPER ET AL.

[l] the Hamiltonian of a supersymmetric quantum mechanical system, with a superalgebra identical to that of Witten’s original N= 1 supersymmetric quantum mechanical model [2]. This supersymmetry was first successfully exploited in the context of the study of chiral anomalies. The reason for this is that the partition function for the supersymmetric quantum mechanics associated with the squared Dirac operator

ZdP) = 1 dx C-4 Wy5epPH) Ix), H= -(0)2 (1.1)

is related in the limit /I= 0 to the chiral U(1) anomaly [3]. Another reason for studying the spectrum of the Dirac operator is the recent

experiments which explore the behavior of QED in the presence of strong external fields [4]. In these experiments conducted at GSI some curious results have been found concerning narrow peaks in the energy of electron-positron pairs coming from heavy-ion scattering. One interpretation of the results of this experiment is that in strong external fields, QED undergoes a phase transition [S].

The starting point for studying QED in strong external fields is the determination of the fermion propagator in an external field. Using the Fock-Nambu-Schwinger proper time formalism [6] one obtains for the fermion propagator in Euclidean space

S(x, x’, A)= (iyD+w~)(--i)[~~ dsexp(-m2s)(xl exp(-Hs) Ix’). (1.2)

The determination of this quantity clearly requires the wavefunctions as well as energy eigenvalues of H. As we will show both Z, and S(x, x’, A) have path integral formulations which are explicitly supersymmetric. We will also show that by introducing fermionic degrees of freedom, one can greatly simplify the determination of the path integral. This path integral representation for S has been studied in a different context by Rajeev who was interested in reformulating quantum electrodynamics as a supersymmetric theory of loops [7].

Using the chiral decomposition of the Dirac operator

b=t(l +~s)fl+t(l-~s)b=Q+ +Q- (1.3)

to obtain a supersymmetric representation of H, Alvarez Gaume [S] constructed a simple proof of the Atiyah-Singer index theorem on compact spaces. This chiral supersymmetry was also exploited by Forgacs et al. to evaluate the partition function and thus the U(1) anomaly for the Dirac operator on both compact and non-compact 2-dimensional manifolds [9]. The calculation of the anomaly proceeds in a fashion identical to that of the determination of the Witten index in SUSY quantum mechanics [lo]. The index is determined by the difference in the density of states of the two partner Hamiltonians. This in turn is related to the derivative of the difference in phase shifts. However, the phase shifts of the partner

SUPERSYMMETRY AND THE DIRAC EQUATION 3

Hamiltonians are related by SUSY, which allows a simple calculation of the anomaly.

In the past people have studied the Dirac equation in particular contigurations without any strategy for finding exact solutions. In a paper written in 1967, Stanciu [ 111 points out that until then, only six configurations of the external field were known to be solvable. He then discusses some new solutions where the problem reduces to a known l-dimensional Schrodinger equation. In his case only one component of the gauge field A was non-zero, and it depended on only one Cartesian coordinate.

Given the renewed interest in the external field problem we have undertaken a systematic study of two different supersymmetric structures of the Dirac equation-the chiral supersymmetry discussed above and another, complex supersymmetry possessed by H [12]. In SUSY quantum mechanics it is now understood that all exactly solvable potentials can be obtained by exploiting the supersymmetry, factorization of the Hamiltonian [13], and a discrete reparametrization invariance-shape invariance [ 143.

The purpose of this paper is to discuss the supersymmetric structure of H and to see how the methods used to obtain analytic solutions for the Schrodinger equation can be extended to finding the eigenfunctions and eigenvalues of H (and therefore also of #). For the case of the Dirac equation in a 3D Coulomb field Sukumar [15] showed how to exploit the supersymmetry along with factorization and “shape invariance” to obtain the complete energy spectrum and eigenfunctions of the Dirac equation. Here we are more interested in the Euclidean Dirac operator.

We will show that all previously found solvable external electromagnetic field configurations can be found by our methods. We also find some configurations which were not known before. We find that in four dimensions the complex supersymmetry leads to a formulation of the solvability problem that is more useful than the chiral supersymmetry.

The rest of the paper is divided as follows: in Section II we discuss the supersymmetric structure of the Dirac equation in 4 dimensions with respect to both the chiral and the complex supersymmetries. We also show how to trivially obtain zero modes using the complex supersymmetry. In Section III we discuss the Dirac equation in two Euclidean space-time dimensions as a preliminary exercise. Because O(4) = SU(2)@SU(2) the 2D solutions are a subset of the solutions to the Euclidean 4-dimensional Dirac equation. We then find all shape invariant potentials in 2 dimensions for which the Dirac equation is exactly solvable. In Section IV we present our strategy for finding solutions for the 4-dimensional Euclidean Dirac operator and find analytic solutions when the superpotential is a function of one variable or the sum of two functions of single variables. In Section V we discuss the path integral formulation of both the partition function and the fermion propagator in an external field and show how to obtain trivially previously known results for the constant external field case. We review in Appendix I how one uses shape invariance and factorization to obtain the exact wavefunctions for the Schriidinger equation. The subset of these solutions relevant to the Dirac problem are given. In

4 COOPERET AL.

Appendix II we show that whenever the l-dimensional Schrodinger equation with potential V(x) is exactly solvable then there always exists a corresponding static scalar field q(x) for which the Jackiw-Rebbi type Dirac problem in 1+ 1 dimensions can be exactly solved. It turns out that q(x) is the superpotential corresponding to the Schrbdinger potential V(x).

II. SUPERSYMMETRIC STRUCTURES OF THE DIRAC OPERATOR IN 4 DIMENSIONS

In 4 dimensions one can show that the square of the Dirac operator possesses two different supersymmetric decompositions. The first of these is the well known chiral supersymmetry.

Defining

Q, = 81 f Ys) YPD,? (2.1)

where

D, = a, - iA,,

one finds that the Dirac operator can be written as

B=Q++Q-. (2.2)

Because Q + and Q _ are nilpotent, Q: = Q2 = 0, the square of the Dirac operator, together with Q + and Q _ yields the usual N = 1 supersymmetry algebra

H= -(Y,D,)~ = {Q, > Q- 1, [Q+,Hl=[Q-,Hl=O (2.3)

first discussed in the context of SUSY quantum mechanics. If we take the following representation of the Euclidean y matrices,

where the 0 are the usual Pauli matrices, then

Y$, = 0

-iD,+a.D iyJ.D]=[;+ ;I

(2.4)

(2.5)

-H=(ypD,)*=D,D,+ a.(B+E)

0 a.(;-E)]=[LoL’ ;L]’ (2-6)

SUPERSYMMETRY AND THE DIRAC EQUATION

Here we have used the convention

Fpv = apA, - &A,, Bi=@kkFjk, Ei = Foi.

We see that if we have

-L+LY,=E,Y,,

then

-LL+(L!P’,)= E,(LY,).

This shows that eigenfunctions of H with energy E, are given by

5

(2.7)

(2.8)

(2.9)

Y= 0 [ 1 and Y= LYl ‘y , [ 1 0

(2.10)

Once we have an eigenfunction of H with a non-zero eigenvalue E, we can easily obtain the eigenfunctions @* of j? with eigenvalues f (El)“’ by the construction

@‘* = C8 f (Ed”‘1 ‘y. (2.11)

In the appendix we also show how to obtain the solutions of the usual massive Dirac equation from Y.

The chiral supersymmetry still requires that we solve the two component equation C(2.8) or (2.9)] to obtain solutions.

When A, has the special form

(2.12)

where the constant 4 by 4 matrix fpy has the properties

f,"= -fvp (2.13)

f,,f,,= -4m? (2.14)

then there exists another breakup of the Dirac operator 8 which leads to the so-called complex supersymmetry. In particular, for all self-dual or anti-self-dual fields the potential can be chosen to have this form. However, (2.12) allows solutions which are not restricted to being dual or anti-self-dual.

The complex supersymmetry relies on the fact that the Dirac matrices can be interpreted as fermionic creation and annihilation operators. This will generalize the form of Q _ and Q, which was found for the SUSY quantum mechanics of the Schriidinger equation. There the matrices cf found in the definition of Q _ and Q + were interpreted as the fermion creation and annihilation operators, tjt and II/.

In SUSY quantum mechanics one has

H=p2+ W2+a,W”= {Q+,Q-}, (2.15)

6 COOPER ET AL.

where

w= f, *+=0-, @=a+ Q, =t,bt(c?,- W)=apex(d,)e-x

Q- = $(a, + W) = o+epx(d,)ex.

(2.16)

The zero energy ground state wavefunctions could therefore be automatically found for any x since for the ground state

Q- !P=Oa Y=epX. (2.17)

The introduction of the ,fermionic degrees of freedom allows a Lagrangian formulation of SUSY quantum mechanics and of the Hamiltonian (fl)’ in which the SUSY transformations can be seen to be bose-fermi transformations. The Lagrangian formulation is exploited in Section V.

To see the complex supersymmetry of the Dirac operator it is simplest to choose a particular basis for the fermion creation and annihilation operators. The general structure of the complex supersymmetry of the Dirac operator in 2D dimensions is discussed in Ref. (12). Let us introduce the two complex variables

u=x,+ix, and 0=x, +ix2, (2.18)

and the fermionic operators

b’ = t(yo k iy, 1, (2.19)

We need the complex derivatives (a, = a/a,,),

and define (the identical equation holds for v)

D, = a, - iA,, B,=a,-&I,.

One can then show that we can write as in (2.2)

fl=m+ +Qu

where now, however, 0 + and &- are defined by

&+ =b,+D,,+b;D,, Qp =b;&+b;d,.

The condition that H = { & + , & ~ 1 is that

&=&=O,

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)


This imposes the integrability condition on the gauge field A,, the solution of which is Eq. (2.12). For our particular decomposition of 4 we have that fPV is given by

[ 2, 21. (2.25)

The Hilbert space breaks up into three subspaces labelled by the fermion number, N, where

(2.26)

The chiral supersymmetry operators Q + and Q _ defined in (2.1) connect the N = 2 plus the N= 0 sector to the N= 1 sector. On the other hand, the complex supersymmetry operator &+ defined by (2.23) takes the zero particle sector into a particular linear combination of the one particle states which is orthogonal to the linear combination obtained by applying Q ~ to a state in the N = 2 sector. These subspaces depend on the superpotential x.

In terms of the superpotential x we can write Q + and & ~ in a manner analogous to Eq. (2.16). That is we have

0, =epx(b,+d,)ex, & = e”(b;&)cX. (2.27)

Here i is summed over the complex coordinates u and u. We will use this form later to find the zero modes of the Dirac operator, just as we found the zero eigenstates of the Schrbdinger equation, in (2.17).

If we analyze the restriction C(2.12) and (2.25)] of having the electromagnetic field A,, derived from a superpotential we find that for our decomposition

E, = -B,, E,= -B,, (2.28)

with E, and B, unrestricted. Another decomposition would have given the result

E,=B,, E,=B, (2.29)

and E, and B, unrestricted. All other decompositions are just 4-dimensional rotations of these two possibilities. We define the states of the Hamiltonian by defining the vacuum to be

b,b, lO,o,) = o. (2.30)

8 COOPER ET AL.

The Hilbert space of solutions decomposes according to the Fermion number operator

s5=8o+S,+fh, (2.31)

and the one particle states also have a natural decomposition for a given superpotential 2,

81 = !jji”’ + sj’,“. (2.32)

The states in sjC,O) are obtained by applying 8, to a zero particle state

Y(,“)=~+(Yo 100)). (2.33)

Since &+ is a fermionic operator, we have

& + Y\“’ = 0. (2.34)

Similarly the states in $ji’) are obtained from

Y”,‘)=&(Y2 11 1)) or & Yy’,2’=0.

In the zero and two particle sectors the Dirac equation reads

-p- &+ Yo=EY(J, -&+& Y2=EY2.

1 (2.35

(2.36 )

If we solve these two equations, then we can reconstruct the whole solution either from the chiral supersymmetry or by using the fact that Yi”) and Yi2) as defined by (2.33) and (2.35) are then solutions in $$i”) and !+jC,*J with the same energy. We have then a twofold degeneracy in the space of solutions as seen from either the chiral or the complex supersymmetry.

The explicit projections of the Dirac operator squared in the zero and two particle sectors are given by

where V is the gradient operator with respect to the ordinary Cartesian variables and f is the 4 x 4 matrix defined by (2.13) and given explicitly for our choice of complex coordinates in (2.25).

If the condition V2x = 0 is satisfied, then the differential equation is identical in the two sectors and we end up with a fourfold degeneracy of all energy eigenvalues. If we take f as given by (2.25) this condition implies that the electromagnetic field is anti-self-dual. In the other inequivalent decomposition of the Dirac operator we would instead be in the self-dual sector when V*x = 0.

One of our strategies to find analytic solutions to the 4-dimensional Dirac equation is to see if there is an additional supersymmetry to relate the solution in the zero particle sector to the solution in the two particle sector.

SUPERSYMMETRY AND THEDIRAC EQUATION 9

In the chiral symmetry language, this is the same as the problem of finding cases where the 2 by 2 matrix Hamiltonian H, = LLt defined in Eq. (2.6) itself contains another SUSY so that it can be written as

H, = [“b” A;+], (2.38)

If we can do that then we can find the solvable external field configurations by knowing solvable quantum mechanical potentials. For example, we can use the technique of factorization and shape invariance to obtain analytic solutions as described in the appendix.

Let us first show how one finds the zero energy solutions. If we assume that these solutions are in the 10 0) sector, then we have the equation

&+K lOO>)=O (2.39)

01

iJu(exY,) 10 1) + 8,(ex!P0) 11 0) = 0.

This can be satisfied only if

ex!Po = f(u, 21) or !Po = epxf(ii, V), (2.40)

where f is an antianalytic function. Thus candidate solutions are of the form

(x0 - i~~)~ (x, - ixz)” e-X. (2.41)

Only those m, n for which the wavefunction is normalizable are bona fide zero mode solutions. If instead the zero modes are in the two particle sector, we obtain, by a similar argument, that the zero modes are of the form

(x0 + ix3)m (x1 + ixl)n ex. (2.42)

If x = x(pl, p2), where pf = xi + x$, p: = x: + xi, then m and n are the conserved angular momenta in the two 2-dimensional planes spanned by (x0, x3) and (x1, x2). This corresponds to the W(2) @ W(2) decomposition of O(4).

Thus we see that when the vector potential A, is derivable from the superpotential x, then the method for obtaining zero modes is quite analogous to what happens in the Schrodinger case.

III. THE SPECTRUM OF THE 2D DIRAC OPERATOR

Although our primary interest is in the 4-dimensional Dirac equation, we first study the 2-dimensional Dirac equation because of its simpler structure and because

10 COOPER ET AL.

one set of solutions to the 4-dimensional equation is just the solution of the 2D equation and another is just a product of two 2-dimensional solutions.

In two dimensions, the chiral and complex supersymmetries are equivalent since we can always by a choice of gauge write

We use the representation

YO=aIY Yl = cJ2. (3.2)

In that representation we have that

(3.3)

where

L=D,-iD,, L+=D,+iD,. (3.4)

We find that (B = For )

(y.D)‘= -(pp-A,)2+.3B (3.5)

from which one immediately reads off LLt and LtL from (3.3). The complex supersymmetry leads to the same result. For the complex super-

symmetry we introduce

b’=a+, d,=$(d,-ia,)

&+ = b,+D,, &- = b,d,, (3.6)

which yield D,, = L, D, = Lt as given above. The complex supersymmetry for (y . D)2 is always ensured in 2 dimensions since

we can always choose the gauge:

In analogy with finding exact solutions to SUSY quantum mechanics we look for potentials that lead to shape invariance. Using (3.7) we find

LL+={V2-V2X-(V~)2+2iVXoEoV} (3.8)

L+L= {V2+V2X-(VX)2+2iVXOsOV}. (3.9)

These equations are similar in structure to those of SUSY quantum mechanics except that we now have a partial differential equation instead of an ordinary differential equation. If we can find a system of orthogonal coordinates and the


superpotential X depends only on one of the two variables, then we can reduce this problem to a l-dimensional quantum mechanical one. For example, in Cartesian coordinates, if we let x = x(x,), then the linear momentum pi is conserved. Letting

!P(x,, x1) = e’p’“‘cp(x,), (3.10)

we find

-LL+= -d;+[p,+Xq2+f’

- L+L = -a; + [p, + x’]’ - x”. (3.11)

This is precisely the structure of SUSY quantum mechanics with the SUSY superpotential

W=p,+x’. (3.12)

For a physical electromagnetic field, x’ cannot depend on pi. Thus only those solvable quantum mechanical potentials of the form W = p, +f(x,,), where f(xO) is independent of pi, lead to solvable Dirac equations with

A I(%) = -f(xd (3.13)

As shown in the appendix, when we solve the shape invariance equation with the above form there are only three solutions: the Morse potential with W(x)= Cl - C2 e ~ ‘-‘, the Rosen-Morse potential with W(x) = C,/C2 + C, tanh(clx), and the oscillator potential W(x) = wx + Cl.

Let us next introduce polar coordinates via x0 = p cos 0, x, = p sin 8. If x = x(p) then we obtain using (3.1)

A,= -PX’, A,=O, (3.14)

where the prime denotes differentiation with respect to p. The electromagnetic field F,, we denote as before as B,

B(p)=&A,-d,A,= -~“-f/p=p~~aA~/ap.

The operators L and Lt which factorize the Hamiltonian take the forms

L=P[a,-ip-l(a,- iA@)],

L+=P[a,+ ip(a,-iA,)].

(3.15)

(3.16)

Following Akhoury and Comtet [lo], we take for the wavefunction

Y= “fiti [ 1 gn ’ fn = p-“*@“.“(p) exp[i(m - t)Q], g, = p-“*@“+‘“(p) exp[i(m + +)I,

where m takes on half integer values.

12 COOPER ET AL.

Then we have that the pairing equations

L!fn=JE,&~ Lgn=Jzfn (3.18)

become

(3.19)

where W = (m - A,)/p. Since we want A, to be independent of m, we see from the appendix that there

are only two shape invariant Schrijdinger potentials that are relevant-namely the free problem in 2-dimensional radial coordinates (with shifted angular momentum) W= (m - @)/p and the 2-dimensional harmonic oscillator W= -iBp + m/p.

The first case is connected with the vortex. Asymptotically the vortex field goes like

A=@V0 (3.20)

so that A, = @, 1= m - @, leading to a shifted angular momentum free problem and corresponding to x = -@ In(p), W= (m - @)/p. A non-singular vortex solution can be obtained [16] by assuming

x= +p’ for p<R

x = -aBR’( 1 + ln(p/R)) for p> R. (3.21)

The second case corresponds to the constant field problem so that

A, = $Bp’, x= -$Bp2. (3.22)

Thus we see that for the 2-dimensional Dirac equation we find only five analytically solvable potentials-three for the case of a conserved linear momentum and two for the case of conserved angular momentum.

IV. THE SPECTRUM OF THE ~-DIMENSIONAL DIRAC EQUATION

The relevant equations that we need to solve in 4 Euclidean dimensions are Eqs. (2.36), where in the N= 0 and N= 2 fermion sectors, $- &+ and 0, & are given by (2.37).

The simplest way of solving the pair of equations (2.36) is to have 2 a function of a single variable. The solutions of Stanciu [ 1 l] arise when we choose that variable to be one of the Cartesian coordinates, say x,. One then chooses for ‘P,, and Y2 the form

(4.1)


One has that A, = -x’(xl) is the only non-zero component of the field and we have

4H,=p~+p:-a:+(PZ+X’)2-X” (4.2)

df?, = pi + p; - 8; + (p2 + 1’)’ + f. (4.3)

We see that H, and H, are SUSY partners with

w= P2 - A,@, 1. (4.4)

Equations (4.2)-(4.3) are essentially the same as (3.11); thus again because we need A, independent of p2 there are only three solvable potentials-the Morse potential with W(x) = C, - C2e-““, the Rosen-Morse potential with W(x) = Cl/C2 + C, tanh(ax), and the oscillator potential W(x) = ox + C,. The eigenvalues and eigenfunctions of these potentials are given in the appendix. These potentials were the ones considered by Stanciu [ 111 in solving the massive Dirac equation. Because we are in Euclidean space, these solutions to the squared Dirac operator in Euclidean space are magnetic field solutions.

The next simplest choice for x is that x=x(p), where p2 = XT +xi for example. Then the magnetic field is just a function of p and we obtain the same result in the (1, 2) plane as in the discussion of the Euclidean 2D Dirac equation. We obtain wavefunctions which are those functions of (p, m) found there times plane waves in the 0 and 3 directions. Thus we have in that case that the only solutions are a constant magnetic field F,, or a vortex field F,*. The solutions are similar to those of the 2D equation with appropriate changes of coordinates.

For x = x(p3), where p: = xf + x2 + xz we get the possibility of Coulomb solutions. In a beautiful paper, Sukumar [ 151 has analyzed in detail how supersymmetry and shape invariance allow one to solve this problem.

The most interesting l-dimensional x is x = X(r), where r* = xi + xf + x: + x:. In that case

2iVx .f .V= -2rp1~‘(r)(A4,, + Ml*), (4.5)

where M, are the O(4) angular momentum generators. Since O(4) = SU(2) @ SU(2) we can simultaneously diagonalize M,,, M12, and M2. Letting the eigenvalues of MO3 = m and M,, = n, M2 = j(j + 2), we find that, after scaling out a factor of r ~ 3’2, the radial equations for H, and H, become

4H,= -~~+r~*[j(j+2)+~]+(f)*-~“+r~‘~‘(2m+2n-3)

4H2= -~~+r-2[j(j+2)+~]+(f)2+f’+r-1,f(2m+2n+3). (4.6)

There are only two solutions where 2 is independent of m, n, j such that Ho and H, are SUSY partners and the potential is shape invariant; namely

x’( r ) = $or (harmonic oscillator), x = or214 (4.7)

(4.8)

14 COOPER ET AL.

which is the constant magnetic field case with

Fp = Q&Y

and f given by (2.25), and a vortex like potential

x’(r) = a/r, X=alnr+C

which leads to the field

A, = af,,-Q

(4.9

(4.10

which as in 2 dimensions corresponds to a shifted 4D angular momentum. As in 2 dimensions one can also solve exactly the case where we combine these two solutions at r = R. So that

X=cor2 for r<R

x = oR2( 1 + In(r/R)) for r2R. (4.11)

The other shape invariant solutions which are possible in the 4D Dirac equation are those for which x is the sum of two functions, each of which is a function of one variable only.

For example, for the particular choice (2.25) forfPy the two 2-dimensional planes (x,,, x3) and (xi, x2) are distinguished. If we choose

x = XOM +x,(x1 13 (4.12)

then p3 and pZ are conserved momenta. Using a product wavefunction of the form

the strutures of H, and H, become

4&J= -J~+(p~+JoXo)2-J~xo-a:+(P2+a,X,)2-d:X1 4H2 = 4 + (P3 + Jc&d2 + J&l - a: + (p2 + J,x*)2 + a;xl.

(4.14)

After separating variables, we get two l-dimensional SUSY Schrodinger equations in the x0 and xi coordinate systems. Thus one can have different shape invariant potentials in say the x,, and x1 directions and still solve analytically the Dirac equation. Each of these equations is exactly the equation we had for the 2-dimensional Dirac equation. Thus we have the restriction again that for 1 to be independent of the conserved pi only those x corresponding to the l-dimensional Morse, oscillator, and Rosen-Morse potentials are exactly solvable.

Similarly because O(4) = SU(2) @ SU(2) we can simultaneously diagonalize say


Mo3, M,,, and M2. Thus we will get a similar structure in the two planes p,, q1 and p2, (p2 if we choose

x = Xl(Pl) + X2(P*L (4.15)

where pT=xz+x:, pz=xf+x,. 2 Taking product solutions we can again separate the equation into two copies of the problem that we had for the 2D Dirac equation. So now we can have following the arguments from Eqs. (3.14)-(3.21) four possibilities with x, and x1 being of the form p* or In p.

This exhausts the solutions we have found for the Dirac equation which relied on our knowing exact solutions to the Schrodinger equation.

V. PATH INTEGRAL FORMULATION OF THE FERMION PROPAGATOR

AND THE WITTEN INDEX

The solvability of the Dirac operator is of course related to the ability to integrate the path integral. In this section we show that by introducing fermionic degrees of freedom, the path integral for the Dirac operator, which is initially a path ordered integral, is reduced to an ordinary path integral. By integrating over the fermionic degrees of freedom, we obtain a purely bosonic path integral. For the case of a constant external field strength F,,, the bosonic path integral is a Gaussian, which allows one to obtain trivially the effective potential for QED as well as the fermion propagator S(x, I, A) and the associated index I.

As stated in the Introduction, the quantities we calculate are the Green’s functions,

G(x, x, t) = (xl e-“’ Ix’) II=‘.

G,(x, x, T) = (xl Tr(y,e-“‘) Is’) IXzi.. (5.1)

By using Schwinger’s proper time formalism we can determine S((x I A) from G. The index Z, is just the spatial integral over G5. Once we introduce fermionic variables then G is determined by choosing antiperiodic boundary conditions for the fermions. If instead we use periodic boundary conditions on the fermions we obtain instead G,. The index of the Dirac operator which is related to the axial anomaly as shown in the papers of Friedan and Windey [l] and Freedman and Cooper [lo] is obtained by integrating G5 over dx as in (1.1).

The square of the Dirac operator is given by

H= (p-eA)* + +pvFp;,,. (5.2)

If we do not introduce auxiliary fermions, then the related matrix valued Lagrangian would be

L(x, ,t) = $t,c?,, + ieA,1, - +apYFp,,

aPy = (2i)-’ [y”, y’] (5.3)

16 COOPER ET AL.

and we would obtain for Feynman’s path integral representation

(xl epHr Ix’) = Bx,(t)Pexp - s [ f

’ dt’ L(x, i) 0 1 , (5.4)

where P denotes path ordering. In analogy with what one does in SUSY quantum mechanics [2], several authors [ 1,7] have notied that one can introduce the Grassman variables tj, via

*, = 2-i’2yll, W,? 4h> = &,.. (5.5)

Then one can write H as

H= (p - eA)2 - &blrFflv$,,. (5.6)

The Lagrangian now becomes

L, = $n,+t, + ieA,i-, + &Q,(t?$,, + F,,)$, (5.7)

which is invariant under the SUSY transformations

6x, = -it+,; s*, = &ip. (5.8)

We now obtain for the path integral

G(x, x; z) = (xl cHr Ix) = f ax,(t) Th,b, exp 1 , (5.9)

where we impose antiperiodic boundary conditions on the fermions at 0 and r. To determine G, we have exactly the same path integral except one imposes periodic boundary conditions on I(/ at t = 0 and r. Since the fermion path integral is quadratic, we can perform the functional integral over the fermionic degrees of freedom exactly for arbitrary F,,.

Consider first the quantum mechanics case where F is replaced by an ordinary function W. After performing the path integral, we must evaluate

where the 1, satisfy

so that

det[a,- W] =fl I,, m

(a,- Wx))~m=4n~m

(5.10)

(5.11)

Y,,, = C, exp s ’ dz’ [A,,, + W(x(z’))]. (5.12) 70


Imposing antiperiodic boundary conditions Y,Jr) = - Y,,JO) yields

A,=z-‘[i(2m+ 1)7r-E(Z)]

E(T) = i,: dz’ W(x(z’))

and the fermion determinant is

L(g)

I- I-

m L(O)

= cash I

’ dz’ ; W(x(z’)), 0

(5.13)

(5.14)

whereas imposing periodic boundary conditions Y*(r) = Y’,(O) yields

A,=zp’[i(2m)n-E(t)]

so that for G, we have that the determinant is given by

(5.15)

rI?, L(g) n m+O Lk=O)

= -fF (l+[$--I’)= -~sinh(~~~dr’I~(x(r~))). (5.16)

To generalize to the case of an antisymmetric matrix F,,,, we can follow the work of Alvarez Gaume in his Bonn Lectures [ 161. Fpy is antisymmetric, so it can be skew- diagonalized:

0

--Xl 0 0

0 0 0

-x2 0 0 -x2 0

(5.17)

In terms of the “eigenvalues” of F, x, we obtain for the fermion path integral

The prime in det’ denotes that we have omitted the zero mode contribution. For antiperiodic boundary conditions the ratio of determinants yields

(5.19)

whereas for periodic boundary conditions relevant to the Witten index we obtain instead

(5.20)

18 COOPER ET AL.

Thus we can always perform the integration over fermions to get a sum of bosonic path integrals for G or G,. Whenever the path integrals can be transformed into Gaussian form, then we can explicitly evaluate the fermion propagator, and the index.

Let us now calculate G and G5 in the constant field case for which

A,, = -+Fp,,xv (5.21)

with Fpy a constant matrix. The Fermion determinant is independent of x and can now be factored out of the path integral. The remaining path integral is Gaussian and can be performed trivially,

2= Dx,(t)exp - I [ i

‘dr’~%,.t,ir~.t,,Fpyxy 0 1

det”/2(d2S,,,/dt2) = det”“( - (d’/dt*)6,, + iF,,,,(d/dt))

(5.22)

In 2 dimensions only F,, = B exists and we can write H in diagonal form,

H=(p-eA)‘+a,B, (5.23)

so that it resembles SUSY quantum mechanics in that

(5.24)

where

H, =(p-eA)‘*B.

Using the above equations we obtain, since x, = B,

Z = B tanh(Br); Z, = B= +E~,,E;~, (5.25)

a result obtained with more difficulty by Akhoury and Comtet [lo] using heat kernel methods and the exact eigenvalues and eigenfunctions.

In 4 dimensions one can use the trick of Schwinger [6] to find the eigenvalues of F. Using the fact that we can write

F,,,F,$ = 86 . PA’ 6 = ;(FF*) (5.26)

Fzv F; + Fpy FYI = 256,, , i-j = a( FF). (5.27)


we find

Thus

x, = [g + (S2- 02)1’2]“2, y = [p(g2LQj’)‘q’/’ .2 (5.28)

For the path integral we therefore obtain

G(x, x; t) = (xl e

which is just the result of Schwinger rewritten in Euclidean space. For G, we obtain

G&x, x; z) = (xl Tr(y,epH’) Ix) = Q = +(FF*). (5.31)

Once we have obtained G, the effective action is obtained by integration,

L(x)=sI dssm’ep”*“G(s). E

(5.29)

(5.30)

(5.32)

This gives an example of the use of the auxiliary fermion variables.

APPENDIX I: SUSY, SHAPE INVARIANCE, AND SOLVABLE POTENTIALS

In this appendix we briefly sketch the procedure for obtaining exact energy eigenvalues, eigenfuctions, and S-matrix elements for a class of Dirac Hamiltonians. There are two separate problems that we are interested in. First, in order to determine field theory quantities such as S(x, y:A) or the index for particular external fields, it is sufficient to know the eigenvalues and eigenfunctions (continuous and discrete) of f12. Then one can directly evaluate Z, and S in Eqs. (1.1) and (1.2) by brute force as was done in two dimensions by Akhoury and Comtet [lo].

Another problem of interest is to find solutions to the massive Dirac equation. Since we are in Euclidean space, we can only discuss magnetic field solutions. As shown by Feynman and Gell-Mann [18] and Brown [19] the solution of the four component Dirac equation in the presence of an external electromagnetic field can be generated from the solution of a two component relativistically invariant equation. The connection is if Y obeys the two component equation,

20 COOPER ET AL.

then the four component spinors that are solutions of the massive Dirac equation are generated from the two component Y via

(4P-eA)+(E-Ao)+mY 1 (ao(p-eA)+(E-A,)-m)Y . (AZ)

For the case that A,=A, =A, =0 and A, =A,(x,) we find that the two component ‘I’ satisfies the equation

[-t3~+p~+(pz-eA,(x,))2+m2+ea,B,]Y=E2~. (A3)

A comparison with Eqs. (4.3) and (4.4) shows that one can determine Y from our solutions to (2.36), (4.3), and (4.4) with the identification

&+m=2E,+p, and En-m=2E,-p,, (A4)

where E, is an eigenvalue of H = -#*, in Eq. (2.36). Once we have reduced the problem to a l-dimensional Schrodinger equation

such as (A3) then one can determine the wavefunctions, eigenvalues, and S-matrix of such an equation using supersymmetry, factorization, and shape invariance.

It is well known that all l-dimensional Schrijdinger Hamiltonians possessing a ground state wavefunction Y0 with energy E. can be factorized [ 13, 141:

H, Y;‘) = [ -d2/dx2 + V,(x, a,)] Yj,‘) = [A+A + Ef ‘1 !P;l’ = E;“Y;‘), (A5)

where

A+ = -d/dx+ W(x, a,), A =d/dx+ W(x, a,)

I’, = W*(x, a,) - dW(x, u,)/dx + E&l’. (A61

The a, are the parameters describing the potential, n labels the states, n = 0, 1, . . . . and n = 0 denotes the ground state. The superpotential W’(x, a,) is given in terms of the ground state wavefunction Yhl)(x, a,) as

W(x, ai) = -d(ln Yh’l(x, u,))/dx. (A7)

For simplicity let us set & (l) = 0. Then the partner Hamiltonian

H, = AA+ = -d’ldx* + V2,

V, = W*(x, a,) + dW(x, u,)/dx (Af3)

gives the same spectrum as H, = AtA but with the ground state missing; that is,

&q’ = 0 Et*’ = E”’ n n+l. (A9)


In addition the eigenfunctions of H, and H, with the same energy are related,

Y(l) (x)= [E;q p* A+Y’*‘(X) II+1 n

!ly(x) = [Ep] -“* A!q11 1(x). (AlO)

It is well understood by now that the degeneracy in the two spectra is due to a supersymmetry [2]. We define the super-Hamiltonian H and the supercharges Q and Qt as

These operators are the 2-dimensional representation of the sl(l/l) superalgebra:

[Q, HI = CQ+, HI = 0, IQ+, Q> = H> {Q, Q} = {Q’, Q+} =O. (A=)

The fact that the supercharge commutes with H gives the energy degeneracy. Clearly this process can be continued. Using factorization, VZ can also be written

as A$A2 + Ei*), and the partner Hamiltonian of this can be constructed. This produces a ladder of potentials, V,. If the partner potentials are “shape invariant” in that V, has the same functional form as Vi but different parameters describing the strength and shape of the potential (except for an additive constant), then as noted by Gendenshtein and Infeld and Hull [14] the full ladder of potentials will be shape invariant,

vrr+ 1(X? %I = V”(X, a,, 1) + C(%), (A13)

and the energy spectrum of the original potential can be determined algebraically:

n+l

EL’)= c C(a,), Ea’ = 0. (A14) k=2

Making use of (AlO) Dutt et al. [14] then showed that the energy eigenfunctions could also be obtained algebraically:

y!,“(x, a,) = fi A+@, a/,) !f$“(x, a,,, 1). (Al51 k=l

In fact by using relations (AlO) and interpreting them as recursion relations for special functions, explicit wavefunctions have been obtained for all known shape invariant potentials [20]. Furthermore, on analytic continuation of these wavefunctions to imaginary values of k, the scattering matrix has also been obtained for these potentials [22].

Let us now come to the question of categorizing the shape invariant potentials, i.e., those which satisfy the condition (A13). This is a highly non-trivial problem as (Al 3) is a non-linear Riccati type of equation. However, a small number of shape

22 COOPER ET AL.

invariant potentials have been found and categorized by assuming that the superpotential W(x, a) factorizes [ 14,211:

ww, 0) = c .t-i(a) g,(x). (A16)

In this paper we have seen that there are basically two different types of relevant l-dimensional problems depending on whether one separates variables in Cartesian coordinates or in polar coordinates. From (2.37) and (3.8) we find that the equation for the square of the Dirac operator is most simple when written in terms of the potential x(x,) which determines the electromagnetic field via (2.12). If x depends on one Cartesian coordinate [I = x(x1), say] then we obtain a l-dimensional quantum mechanical problem where the superpotential has the form [see (3.4) (4.3), and (4.4)]

Wx,)=P,-AAx1). (.417)

If x is the sum of two functions of one variable of the form [see (4.14)]

x = x(x, ) + x(x3) (A181

then one can separate variables and one has a l-dimensional problem in both the 1 and 3 directions with W having the form of Eq. (A17) (with different variables xi, etc.).

For l-dimensional problems having the form

W(x) = p; - Ai (A19)

one needs that the electromagnetic field Ai is independent of the conserved linear momentum pi. This puts severe constraints on the acceptable solutions of the shape invariance condition:

W2(x, al) + w’(x, a,) = W2(x, a()) - w’(x, a()) + C(q)). (A20)

If we let W(x, p) = p + f(x) we obtain the equation for f(x) (a, = p, a, = p - CC),

2f’(x)-2af= p* - (p-cry- C(p). (A.21 1

In order forf(x) to be independent of p, the r.h.s. of (A21 ) must be independent of p. Thus we choose

C(p2) = p2 - (p-a)’ - 2k (i-422)

and obtain

f(x)=k/a+be-“” (~23)


with k, ~1, and b arbitrary parameters. This gives the Morse potential:

W(x)=p,+C,+C,cl~‘(l-ePV). (~24)

For the special case where CI = 0 we obtain instead the oscillator potential

W(x) = p, + wx + c. (425)

The other way of obtaining a solution independent of p is to assume that

W(x) = B/A + Af(x), (A’W

where B/A = p, and we want A and f(x) independent of p. Now we choose a, = A, a, = A - LY under our shape invariance condition, and we find that the solution is

f(x) = tanh(ctx), 6427)

giving rise to the Rosen-Morse potential

W(x) = p + A tanh(ax) + C. (A-28)

The next type of problem was that where the potential was of the form x = I or x = I + x(p3), where p: = XT + xz, p: = xi + xi. In that case we found that the problem reduced to knowing the solution to the 2-dimensional Euclidean Dirac problem (see (3.19)) with W(p) having the form

6429)

From the shape invariance equation with a,, = m, a, = m + 1, we obtain the differential equation

dr+LC 4 P 2'

(A301

In order for A, to be independent of m we need C to be independent of m. The general solution is

fq-;, (A311

where c1 is an integration constant. For CI = 0 we obtain the 3-dimensional oscillator potential with C = 2w,

W(p) = mlp + imp, (~32)

24 COOPER ET AL.

which corresponds to a constant magnetic field. For C= 0 we obtain instead a shifted angular momentum free theory

W(P) = (m + cc)lP> (A33)

which corresponds to a vortex solution. In 4-dimensional problems with spherical symmetry we found (see (4.6)) that W

was of the form

W( r ) = a/r + x’. (A34)

This is exactly the same form as that in 2 dimensions (except p is replaced by r) so that the same analysis applies.

We now tabulate the eigenfunctions and the energy eigenvalues associated with these particular shape invariant potentials using Eqs. (AlO), (A14), (A15). The method of obtaining these results is discussed in detail in Refs. [ 14,20,21]:

i. Morse Potential.

W(x) = A - Be-*‘; V~(x)=AZ+B2e~*““-2B(A+a)e-““;

E, = A2 - (A - ncc)*;

ul,(y) = y”p”epY/2~~-2n(y); y=2Be-mx, A S=--. CI ci

(A35)

ii. Rosen-Morse Potential.

W(x) = A tanh LXX + B/A; V_(x) = A* + B*/A* + 2B tanh ux - A(A + a) sech*(crx);

E, = A2 - (A - ncr )’ + B*/A* - B*/( A - ncr )*;

Yn(y)= (1 _ y)(s-n+m (1 + y)+n-a)/2 p;Pn+u.s-n-a)(y);

y = tanh KY, s = A/a, A= B/CC’, a = ,I/(s - n).

iii. Shifted 1D Oscillator.

W(x) = &ox - 6; V/_(x) = fo’(x - 2b/w)2 - +o;

E, = nw;

Yn( y) = ed’2Hn( y); y=&(x-;).

iv. 3-Dimensional Oscillator.

W(r)=&or-(l+ 1)/r; T(r) = fw2r2 + l(l+ 1 )/r2 - (I+ 4)~;

E, = 2nw;

y =&or*.

(A36)

(A37)

(A3g)


v. Shifted Angular Momentum (Bessel Functions).

In this case there are no bound states and C(m) is zero. The equation

2” - (m* - a)Z/r’ + IZ = 0 (A39)

is factorized with

W(r, m) = (m -+)/r. (A40)

The operators A and At, as noted by Infeld and Hull [ 141, give the two recurrence formulas for the Bessel functions,

Z m+l =A-“‘{(m+f)/r-d/dr} Z,

Z,=E.~“*{(m+f)/r+d/dr}Z,+,,

from which we realize

Z, = r’i2J,,,(L”2r).

(A41 )

Given the above wavefunctions we are now in a position to obtain the eigenfunctions and eigenvalues of H = -#* as well as the solutions of the massive Dirac equation.

For example, to solve Eqs. (2.36) with W given by (4.5) and a Rosen-Morse potential,

A, = -eH,a-’ tanh(ax,), (A43)

one has that the wavefunction corresponding to the eigenfunction with eigenvalue one has that the wavefunction corresponding to the eigenfunction with eigenvalue A,, = 4E, - p$ - pi in the 5O sector is A,, = 4E, - p$ - pi in the 5O sector is

ei(mw + ~2.~2 + P~.T~)Q,(~(~~)) ; 1, (A44)

where Gn and y are given in (A36) with the identification that

A = eH,,/ol, B = p2 eH& 2, = A2 - (A - nc1)2. (A45)

In the b2 sector a similar wavefunction having the same eigenvalue 1, is obtained, being non-zero only in the first column of the four component spinor, which because it is the SUSY partner of the previous wavefunction has A replaced by A - c1= eHO/cr -CC. To obtain the other two solutions with the same I,, one can use

26 COOPER ETAL.

either the complex SUSY operators Q, and Eqs. (2.23) and (2.25) or the chiral SUSY operator Q, (see (2.1) and (2.5)),

1 (A46)

to reconstruct the other two solutions. If instead we want solutions to the massive Dirac equation we consider instead

the two component Y defined by

[

0 'yb"'= ,~(Po.~o+P?.~?+P3.~3)~n(Y(X,)) 1 (~447)

as input in Eq. (A2) and uses,the identification of (A4).

APPENDIX II: EXACT SOLUTIONS OF THE (I+ 1 )-DIMENSIONAL DIRAC EQUATION WITH A SCALAR FIELD

In this appendix we sketch the procedure for obtaining exact energy eigenvalues, eigenfunctions, and hence the S-matrix for the Dirac equation involving a scalar field in 1 + 1 dimensions. These models are useful in the context of the phenomenon of fermion number fractionalization which has ben seen in certain polymers like polyacetylene.

For these systems the Dirac Lagrangian is given by

L= i!P(x)y%yP(x)- qx)(p(x) F(x), (Bl)

where q(x) is the static, finite energy solution corresponding to the scalar field Lagrangian

~,=;a,~ ap'cp- qcp). WI

Note that in (Bl) and (B2) we have absorbed the coupling constant in cp and V(q), respectively. We would now like to address the question What are the various forms of the scalar field q(x) for which the Dirac equation corresponding to (Bl) can be exactly solved? We will show now that whenever the l-dimensional Schriidinger equation is exactly solvable for a given potential V(x) then there always exists a corresponding scalar field q(x) for which the Dirac equation is also exactly solvable.

The Dirac equation following from (Bl ) is

ifa,q~,t)-cp(~) ~(x,t)=o.

Let

Yqx, t) = e-iu’Y(x)

(B3)

(B4)

SUPERSYMMETRYANDTHEDIRAC EQUATION 27

so that (B3) reduces to

yOwY(x) + iy’ dY(x)/d.x - q(x) Y(x) = 0.

Following Jackiw and Rebbi [23] we choose

p = 0.1, yl = ia3, Y(x) = Y,(.x) [ 1 Y2(-u)

W)

WI

so that we have the coupled equations

Y\(x) + q(x) Y,(x) = COY&)

Y;(x) - q(x) Yz(x) = -wYy,(x).

W’a)

W’b)

These equations are easily decoupled yielding

- Y’;(x) + [q2(x) - cp’(x)] Yu,(x) = 02Yy,(x) (B8a)

- (u:(x) + [q*(x) + cp’(x)] Yu,(x) =w*w4 Wb)

These are precisely the Schrodinger like equations corresponding to the supersymmetric partner potentials

V*(x) = q’(x) f q’(x). (B9)

As shown in Ref. (21), not only the shape invariant potentials but also the entire class of solvable Natanzon potentials (these are potentials whose wavefunctions are hypergeometric and confluent hypergeometric functions) can be cast into the SUSY form (B9), with q(x) being related to the ground state wavefunction of the related quantum mechanics problem via (A7) [here W is replaced by cp]. These potentials have wavefunctions and spectra that can be determined algebraically by exploiting supersymmetry, shape invariance, and hidden shape invariance [ 14,211. Using the construction given above one can then immediately find the solution of the corresponding Dirac Lagrangian (B 1).

ACKNOWLEDGMENTS

F.C. thanks Alan Chodos for suggesting this study and for discussions. R.M. thanks LANL for its warm hospitality. This work was supported in part by the DOE and by the Italian M.P.I.

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Documents

Supersymmetry and the Dirac Equation...Euclidean 4-dimensional Dirac equation. We then find all shape invariant poten- tials in 2 dimensions for which the Dirac equation is exactly